A Width-Range Boundary Atlas for Fixed-Point Quantum LDPC Min-Sum Decoding

Technical Summary: A Width–Range Boundary Atlas for Fixed-Point Quantum LDPC Min-Sum Decoding

Problem Statement
Practical quantum low-density parity-check (qLDPC) decoders must operate in finite precision. The literature frequently characterizes these decoders simply by word width (e.g., "8-bit"), implicitly assuming that the number of bits alone determines whether a format is viable. However, a fixed-point format is defined by two quantities: word width ($w$) and a clip level ($L$) that bounds representable magnitude. These two parameters trade off directly: widening the range reduces clipping but coarsens resolution, while narrowing the range does the reverse. The paper argues that width is an insufficient specification because the decoder's operating point depends on where the message-magnitude distribution sits relative to the clip level, a relationship further modulated by the normalisation gain ($\alpha$) in normalised min-sum decoding. The study seeks to determine whether the performance and resource efficiency of fixed-width formats remain consistent across different codes, Calderbank–Shor–Steane (CSS) sectors, physical error rates, and normalisation recipes.

Methodology
The study is a preregistered descriptive investigation conducted in three staged phases. The endpoint is the logical error rate (LER) under a code-capacity model, compared against a matched floating-point reference.

• Experimental Bank: The study evaluates 19 feasible fixed-point formats across 48 frozen cells. The cells are formed by the Cartesian product of:
  • Codes: Four public codes (one Hypergraph-Product [HGP] and three Bivariate-Bicycle [BB] codes).
  • Sectors: Both X and Z sectors of the CSS construction.
  • Error Rates: Three physical depolarising error rates ($p \in \{0.015, 0.020, 0.025\}$).
  • Recipes: Two normalisation recipes (a "per-code" recipe fitted in prior work and a "common" recipe with $\alpha=0.75, \beta=0$).
• Formats: Formats are indexed as $ww\_rr$ (word width $w \in \{5,6,7,8\}$ and clip level $r \in \{2,4,8,16,32\}$). The study uses signed two's-complement arithmetic with round-half-away-from-zero and saturating overflow.
• Decision Rules:
  • Non-inferiority: A format is non-inferior if the upper bound of the 95% bootstrap confidence interval for the paired LER ratio (fixed-point vs. floating-point) is $\le 1.10$.
  • Resource Qualification: A format meets the joint target if it is non-inferior AND achieves at least a 25% reduction in message traffic (bits moved per frame) relative to an $8\text{-bit}, \text{clip}=8$ baseline.
  • Unresolved Cells: Cells with fewer than 150 discordant pairs (where fixed-point and floating-point decoders disagree) are marked "unresolved" regardless of the point estimate, as they lack statistical power to declare non-inferiority or inferiority.
• Controls: The study includes plain min-sum (unnormalised) controls to isolate the effect of the normalisation recipe.

Key Results

1. No Universal Winner: No single fixed-point format was jointly resource-qualified and non-inferior in all 48 cells. The set of formats meeting the joint target in every cell is empty.
2. Width is Insufficient: Holding word width constant does not guarantee consistent performance.
   • At 8 bits, the format $w8\_r32$ (clip level 32) was non-inferior in 16 of 48 cells and inferior in none.
   • Conversely, $w8\_r2$ (clip level 2) was inferior in all 48 cells.
   • This demonstrates that "8-bit" is not a sufficient specification; the clip level and normalisation recipe are critical.
3. Clip Level Dominance: Clip level 2 was inferior in all 48 cells across all widths. The favourable region is confined to wider clip levels ($L \ge 16$), though large areas remain unresolved.
4. Trade-offs and Leaders:
   • The format $w5\_r16$ achieved the highest count of joint-target successes (10 of 48 cells) with a mean traffic saving of 30.6%.
   • However, $w5\_r16$ was inferior in 10 cells, all under the "per-code" recipe. Under the "common" recipe, it had zero inferior cells.
   • Wider formats (e.g., $w8\_r32$) often failed the resource condition (traffic saving) not because of decoding quality, but because they did not converge faster than the baseline to offset their wider bit-width.
5. Unresolved Data: A significant portion of the atlas (420 of 912 format-cell combinations) is unresolved. Section 4.9 clarifies that most of these (e.g., 25 of 28 for $w5\_r16$) are unresolved due to falling below the discordance floor (too few disagreements with the reference), rather than having confidence intervals that straddle the margin. This implies the study cannot certify "safe" formats that behave nearly identically to floating-point, only those that fail measurably.
6. Normalisation Sensitivity: The choice of normalisation recipe ($\alpha, \beta$) significantly alters verdicts. For instance, $w5\_r16$ was inferior in 10 cells under the per-code recipe but had no inferior cells under the common recipe.

Significance and Claims
The paper claims to provide a "boundary atlas" that separates format failures from limits of statistical power. Its primary contribution is the direct evidence that word width is not a sufficient specification for fixed-point qLDPC min-sum decoding. The operating point is determined by the combination of word width, the clip level, and the normalisation recipe.

The author is modest about the scope of their claims:

• The results are descriptive, not confirmatory. No held-out confirmation was run (declined at preflight due to projected marginality), and no random-effects model was used to claim generalizability across code families.
• The findings apply to the specific "frozen bank" of four codes and the specific code-capacity regime tested (LERs between $6 \times 10^{-4}$ and $2.2 \times 10^{-2}$). The author explicitly states that rankings should not be extrapolated to regimes with much lower error rates ($<10^{-4}$) where precision behavior may change.
• The study does not claim to identify the "optimal" format, as the grid does not bracket the optimum (e.g., it stops at $L=32$) and the resource threshold (25%) is a preregistered point on a curve, not a universal deployment requirement.
• The study highlights that "unresolved" outcomes are often a lack of statistical power to distinguish near-identical performance, rather than evidence of safety.

In conclusion, the paper argues that reproducible finite-precision reports must include width, the normalisation recipe, and the clip level, as width alone fails to capture the decoder's operating point or its resource-performance trade-offs.